Interpretation Panel fixed effect with log dependent variable

[Anne Klein]

Dear Stata users,

I am running a fixed effect model on household data with log household per capita consumption as dependent variable and a binary shock indicator as independent variable.

xtreg log_consumptionpc shockindicator controls, fe with t=1, 2

My data is a two-period panel with every household observed in t=1 and t=2. Therefore, the fixed effect model is equivalent to a first-difference model. The fixed effect is a household fixed effect. Is it correct that the obtained coefficient for the binary shock variable in the fixed effect model with log dependent variable then captures the change in the change of consumption due to the shock? I do not really understand how to interpret the coefficient of a variable in a two-period panel Regression with Household fixed effect. Maybe you can help with that?
Best, AC

[Carlo Lazzaro]

Anne:
hopefully a toy-example can shed some light on the issue you're intreseted in:

. use "https://www.stata-press.com/data/r16/nlswork.dta"
(National Longitudinal Survey.  Young Women 14-26 years of age in 1968)

. xtreg ln_wage i.year c.age##c.age if year<=69, fe

Fixed-effects (within) regression               Number of obs     =      2,598
Group variable: idcode                          Number of groups  =      1,754

R-sq:                                           Obs per group:
     within  = 0.1140                                         min =          1
     between = 0.0948                                         avg =        1.5
     overall = 0.0986                                         max =          2

                                                F(3,841)          =      36.08
corr(u_i, Xb)  = -0.0383                        Prob > F          =     0.0000

------------------------------------------------------------------------------
     ln_wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     69.year |   .0497484    .042309     1.18   0.240    -.0332952     .132792
         age |   .2609896   .1146971     2.28   0.023     .0358634    .4861158
             |
 c.age#c.age |  -.0048071   .0025014    -1.92   0.055    -.0097167    .0001026
             |
       _cons |   -1.88574   1.432257    -1.32   0.188    -4.696958    .9254784
-------------+----------------------------------------------------------------
     sigma_u |   .3629493
     sigma_e |  .20723179
         rho |  .75414644   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(1753, 841) = 4.12                   Prob > F = 0.0000


. di (exp(.0497484)-1)*100
5.100663

.

If you have a logged dependent variable, each regressioin coefficient expresses (when adjusted for the othere ones) the variation (in percentage) of the conditional mean of the regressand.
In the toy-example reported above, other things being equal, switching from year 68 to year 69 increase the conditional men of -ln_wage- of about 5,10% (let's forget for a while that the related p-value is clearly unsignificant and that, as such, there's no evidence that -69.year- coefficient differs from zero as confirmed by -test-):

. test 69.year

 ( 1)  69.year = 0

       F(  1,   841) =    1.38
            Prob > F =    0.2400

.

As an aside, you could find Paul Allison's (laudably brief) textbook https://us.sagepub.com/en-us/nam/fix...els/book226025 interesting.

Eventually, I would add in your code -i.year-.

[Anne Klein]

So it‘the the conditional mean for each young women, right? Since the fe is a household fixed effect. And it‘s the conditional mean because it‘a conditional for example in my case on facing a shock, i.e. when the shock variable turns 1.

[Carlo Lazzaro]

Anne:
all the -e(sample)- observations contribute to the conditional mean of the dependent variable (in your case, all the households included in the regression contributes to the conditional mean), that is conditional on the predictors included in the right-hand side of the regression equation (different sets of predictors=different conditional means).